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**Department of Mathematics**

**UNIVERSITY OF ROCHESTER**

**Tel (716) 275-4429 or 244-9368**

**FAX (716) 244 6631 (at U of R)**

**email: RARM@math.rochester.edu**

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*Ralph
A. Raimi*

*Professor
Emeritus*

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To the Editors,

__Journal for Research in
Mathematics Education__:

1906 Association Drive

Reston, VA 20191-9988

__Mathematicians Writing__, by Leone Burton and Candia
Morgan, in volume 31 of JRME (July, 2000) pp 429-453, contains some errors
that should be noted in JRME, and amended before their interesting research is
pushed much further. I refer in particular
to the __Indicators of Authority__ that came under discussion on page
439-440.

"One important aspect of the identities of authors as
projected by their texts is the extent to which and the manner in which they
claim to be authoritative within their community.....

"Terms such as __clearly__ and __obvious__ …
[serve] as a claim to authority on the part of the writer (implying that this
derivation is clear to me and I do not need to explain it further because if
it is not clear to you, that is your fault not mine) ...... We believe that,
whatever the author's intent, the extent or absence of such words is one of the
interpersonal aspects of the writing that will influence the ways in which
the readers of the text will construct an image of the author and will consequently
judge the worth of the text itself."

Now it might be that -- "whatever the author's __intent__"
might be -- some readers might judge the character of the author and the worth
of his work under the influence of having seen terms such as __obvious__ in
that work; but to say -- as Burton and Morgan clearly do in the immediately
preceding sentences -- that using such a term constitutes a claim to __authority__
is not correct.

It is by authority that I know that Napoleon was defeated
at Waterloo in 1815, as I have no firsthand knowledge of this alleged fact,
only the statements (albeit rich and interlocking) of persons I believe. We all make use of authority in this way
every minute of our lives, and give it little thought. But in the writing of
mathematical papers, there is never an
appeal to authority.

Mathematical statements are almost unique in their failure
to appeal to authority. If there is a gap in an argument, the word __obvious__
does the very opposite of concealing the gap, and is not asking the reader to
accept the writer's word on the matter.
__Obvious__ and __clearly__ merely abbreviate the discussion for
the benefit of the persons who already know the connecting link that is being
omitted.

Of course it is a matter of judgment for the author to
decide what parts of his argument may be omitted for expository purposes, and
some authors presuppose more skill or previous knowledge on the part of the
reader than others do. Sometimes the
assessment of audience fails; it has often been an occasion of merriment among
mathematicians when someone (in print or at the blackboard) imagines obvious
that which is not -- not even, on reflection, to the author himself. But to imagine that the author is expecting
his reputation to overwhelm the reader by using such phrases for intimidation
is simply mistaken. There is no form of
literature less "authoritative" than mathematics.

On the following page, a few sentences later, Burton and
Morgan write,

"Indicators of authority claims from the use of
modifiers, such as __clearly__, __easily__, __of course__, and __immediately
obvious__, and phrases, such as __without loss of generality__, __it
suffices to consider the case__, and __the last stage is trivial__, all
signal a gap in the argument (implying that there is no accompanying justification
for the claim that generality is not lost) and hence signal to the readers that
they should accept the author's claim."

There are many things that can be said about the use of
such phrases to abbreviate a mathematical discussion, but that they
"signal to the readers that they should accept the author's claim" __on
the author's authority__ is not one of the correct ones.

"Without loss
of generality", for example, is a technical phrase used in consideration
for the reader (and publisher). To consider
it an assertion of authority rather than a part of the proof is analogous to
considering "Let x be an element of G" an intrusion on the reader's
liberty to consider x in other ways; no mathematician would imagine that the
apparently peremptory tone of "Let x be an element of G" constitutes
a claim to authority.

An explanation (by example) of __without loss of generality__
might be of value here, for it would surely not be wise to ask the present
reader to accept my preceding paragraph on my own authority, either: In proving by use of a coordinate system the
theorem that the medians of a triangle meet at a point, a mathematician might
begin by saying, "Let ABC be the triangle. Without loss of generality, suppose A is at the origin (0,0) and
AB is along the x-axis with B at the point (b,0)."

It is thus asserted, in other words, that proving the
theorem for this specially placed triangle proves it for all other
triangles. Had the proof begun by
asking, in addition, that the point C be given the coordinates (b,b), the
mathematician could be called to order for inappropriate use of
"WOLOG" (a common informal abbreviation for __without loss of
generality__, so frequent and so little idiosyncratic is the use of the
term), for the triangle thus specified would a right triangle, and proving
things for a right triangle is not proving it for all triangles.

Why, on the contrary, does placing two of the vertices at
(0,0) and (0,b) fail to lose generality?
Surely not every triangle in the plane occupies so particular a
position. Isn't this mathematician
cheating a bit, making it easier for himself by this ploy, intimidating his
less secure readers by some claim to authority?

Well, it is the genius of the entire method of coordinates,
that the Euclidean properties of Euclidean figures remain invariant under
rigid motions, that concurrence and distances are Euclidean properties in this
sense, and that there is always a rigid motion which will bring any triangle,
wherever placed initially, to the position described in this proof.

Now the whole story of how Euclidean geometry can be got at
via coordinate systems is really quite long
Not everyone has learned these things, and a high school student using
analytic geometry for the first time often does take such missing links on
faith, sometimes not even noticing that anything is missing. If the mathematician were writing for that
high school student he might well avoid "WOLOG" and provide a longer
explanation. Yet mathematical and
scientific exposition would come to a stop if all papers began "at the
beginning." Mathematical research
papers, especially, are father from first principles than most expositions
because directed at professional mathematicians.

On the other hand, the path back, from where the expositor
is now, to where the first principles are to be found, must always be
visible. In tackling this theorem
about the centroid of a triangle my imagined author is presuming the reader
has already come a certain distance in analytic geometry. Yet rather than merely placing the triangle's
base on the x-axis with a vertex at the origin, this particular exposition
makes a point of __reminding__ the reader, via "WOLOG", that there
is no sleight-of-hand at work.

Far from being an appeal to authority, the phrase __without
loss of generality__ is a reminder that some "first principles,"
or some of the intervening logical paths, are being omitted, so that the
uncomprehending reader, who might otherwise think himself obtuse for not
seeing why the proof for the special case proves them all, is given the more
pleasant information that he is merely missing some information. At the same time, the __comprehending __reader,
the author's expected audience, is saved some tedium. So with the other phrases cited by Burton and Morgan as
assertions of authority. Take it from
me, they are no such thing.

Sincerely yours,

Ralph A. Raimi